BETON PRATEGANG

Members Analysis under Flexure (Part II)

IntroductionThe analysis of flexural members under service loads involves the calculation of the following quantities: Cracking moment. Location of kern points. Location of pressure line.

Cracking MomentThe cracking moment (Mcr) is defined as the moment due to external loads at which the first crack occurs in a prestressed flexural member. Considering the variability in stress at the occurrence of the first crack, the evaluated cracking moment is an estimation and important in the analysis of prestressed members.

Cracking Moment

(contd)

Based on the allowable tensile stress the prestress members are classified into three types as: 1. Type 1: Full Prestressing, when the level of prestressing is such that no tensile stress is allowed in concrete under service loads. 2. Type 2: Limited Prestressing, when the level of prestressing is such that the tensile stress under service loads is within the cracking stress of concrete. 3. Type 3: Partial Prestressing, when the level of prestressing is such that under tensile stresses due to service loads, the crack width is within the allowable limit.

Cracking Moment

(contd)

For Type 1 and Type 2 members, cracking is not allowed under service loads. Hence, it is imperative to check that the cracking moment is greater than the moment due to service loads. This is satisfied when the stress at the edge due to service loads is less than the modulus of rupture.

Cracking Moment

(contd)

The modulus of rupture is the stress at the bottom edge of a simply supported beam corresponding to the cracking moment (Mcr). The modulus of rupture is a measure of the flexural tensile strength of concrete and measured by testing beams under 4 point loading including the reaction. The modulus of rupture (fcr) is expressed in terms of the characteristic compressive strength (fck) of concrete in N/mm2.(Eq. 13)

Cracking Moment

(contd)

FIg. 9 shows the internal forces and the resultant stress profile at the instant of cracking.

Fig. 9 Internal forces and resultant stress profile at cracking

Cracking Moment

(contd)

The stress at the edge can be calculated based on the stress concept and the cracking moment (Mcr) can be evaluated by transposing the terms as Eq. 14. This equation expresses Mcr in terms of the section and material properties and prestressing variables.

(Eq. 14)

Kern PointsWhen the resultant compression (C) is located within a specific zone of a section of a beam, tensile stresses are not generated. This zone is called the kern zone of a section. For a section symmetric about a vertical axis, the kern zone is within the levels of the upper and lower kern points. When the resultant compression (C ) under service loads is located at the upper kern point, the stress at the bottom edge is zero. Similarly, when C at transfer of prestress is located at the bottom kern point, the stress at the upper edge is zero. The levels of the upper and lower kern points from CGC are denoted as kt and kb, respectively.

Kern Points

(contd)

Based on the stress concept, the stress at the bottom edge corresponding to C at the upper kern point , is equated to zero. Fig. 10 shows the location of C and the resultant stress profile when compression is at upper kern point.

Fig. 10 Resultant stress profile (compression at upper kern point)

Kern Points

(contd)

The value of kt can be calculated by equating the stress at the bottom to zero as follows:

(Eq. 15)

Eq. 15 expresses the location of upper kern point (kt) in terms of the section properties. Here, r is the radius of gyration and yb is the distance of the bottom edge from CGC.

Kern Points

(contd)

Similar to the calculation of kt, the location of the bottom kern point, kb can be calculated by equating the stress at the top edge to zero. Fig. 11 shows the location of C and the resultant stress profile when compression is at lower kern point.

Fig. 11 Resultant stress profile (compression at lower kern point)

Kern Points

(contd)

The value of kt can be calculated by equating the stress at the top to zero as follows:

(Eq. 16)

Eq. 16 expresses the location of lower kern point (kb) and yt is the distance of the top edge from CGC.

Cracking Moment using Kern Points

The kern points can be used to determine the cracking moment (Mcr). The cracking moment is slightly greater than the moment causing zero stress at the bottom. C is located above kt to cause a tensile stress fcr at the bottom. The incremental moment is fcrI/yb. Fig. 12 shows the shift in C outside the kern to cause cracking and the corresponding stress profiles.

Cracking Moment using Kern Points (contd)

Fig. 12 Resultant stress profile at cracking of the bottom edge

Cracking Moment using Kern Points (contd)

The cracking moment can be expressed as the product of the compression and the lever arm. The lever arm is the sum of the eccentricity of the CGS (e) and the eccentricity of the compression (ec). The later is the sum of kt and z, the shift of C outside the kern.

(Eq. 17)

Cracking Moment using Kern Points (contd)

Pressure LinesThe pressure line in a beam is the locus of the resultant compression (C) along the length. It is also called the thrust line or C-line. It is used to check whether C at transfer and under service loads is falling within the kern zone of the section. The eccentricity of the pressure line (ec) from CGC should be less than kb or kt to ensure C in the kern zone.

Pressure Lines

(contd)

The pressure line can be located from the lever arm (z) and eccentricity of CGS (e) as follows. The lever arm is the distance by which C shifts away from T due to the moment. Subtracting e from z provides the eccentricity of C (ec) with respect to CGC. The variation of ec along length of the beam provides the pressure line.

(Eq. 19)

Pressure Lines

(contd)

A positive value of ec implies that C acts above the CGC and vice-versa. If ec is negative and the numerical value is greater than kb (that is |ec| > kb), C lies below the lower kern point and tension is generated at the top of the member. If ec > kt, then C lies above the upper kern point and tension is generated at the bottom of the member.

Pressure Lines

(contd)

Pressure Line at Transfer The pressure line is calculated from the moment due to the self weight. Fig. 13 shows that the pressure line for a simply supported beam gets shifted from the CGS with increasing moment towards the centre of the span.

Fig. 13 Pressure line at transfer

Pressure Lines

(contd)

Pressure Line under Service Loads The pressure line is calculated from the moment due to the service loads. Fig. 14 shows that the pressure line for a simply supported beam gets further shifted from the CGS at the centre of the span with increased moment under service condition.

Fig. 14 Pressure line under service loads

Pressure Lines

(contd)

Limiting Zone For fully prestressed members (Type 1), tension is not allowed under service conditions. If tension is also not allowed at transfer, C always lies within the kern zone. The limiting zone is defined as the zone for placing the CGS of the tendons such that C always lies within the kern zone.

Pressure Lines

(contd)

For limited and partially prestressed members (Type 2 and Type 3), tension is allowed at transfer and under service conditions. The limiting zone is defined as the zone for placing the CGS such that the tensile stresses in the extreme edges are within the allowable values.

Fig. 15 Limiting zone for a simply supported beam

Example 1

For the post-tensioned beam with a flanged section as shown, the profile of CGS is parabolic, with no eccentricity at the ends. The live load moment due to service loads at mid-span (MLL) is 648 kNm. The prestress after transfer (P0) is 1600 kN. Grade of concrete is 30 MPa. Assume 15% loss at service.

QuestionEvaluate the following quantities: a. Kern levels b. Cracking moment c. Location of pressure line at mid-span at transfer and at service. d. The stresses at the top and bottom fibres at transfer and at service. Compare the stresses with the following allowable stresses at transfer and at service. For compression, fcc = 18.0 N/mm2 For tension, fct = 1.5 N/mm2

Moment of cracking (Mcr):

Since the given live load moment (648.0 kNm) is less than the above value, the section is uncracked the moment of inertia of the gross section can be used for computation of stresses.

Solution(contd) c. Calculation of location of pressure line at mid-span At transfer Since ec is negative, the pressure line is below CGC. Since the magnitude of ec is greater than kb, there is tension at the top.

Solution(contd)

At transfer

Solution(contd) At service Since ec is positive, the pressure line is above CGC. Since the magnitude of ec is greater than kt, there is tension at the bottom.

Solution(contd)

At service

Solution(contd) d. Calculation of stresses The stress is given as follows:

Solution(contd)

Calculation of stresses at transfer (P = P0):

Solution(contd)

Stress at the top fibre (at transfer):

Solution(contd)

Stress at the bottom fibre (at transfer):

Solution(contd)

Calculation of stresses at service (P = Pe):

Solution(contd)

Stress at the top fibre (at service):

Solution(contd)

Stress at the bottom fibre (at service):

Solution(contd) The stress profiles:

The allowable stresses are as follows: For compression, fcc = 18.0 N/mm2 For tension, fct = 1.5 N/mm2. Thus, the stresses are within the allowable limits.